I'm not familiar with Mojo::DOM but this is how I would do it using HTML::TreeBuilder::XPath. If you don't know them already you will need to learn some Xpath expressions, but that is a useful thing to know.

Note that <i class="fa fa fa-phone"></i> does not actually contain the phone number - it is a FontAwesome element that displays an icon - so you need to go up one level to get the content: $phone_icon->parent->as_text. Same goes for all the other FontAwesome elements.

You may need to add checks for elements that are missing in some members. PS: I had to fix the image source in your example.

use Data::Dumper;
use HTML::TreeBuilder::XPath;

my $data = join '', <DATA>;
my $tree = HTML::TreeBuilder::XPath->new;
$tree->parse($data);
$tree->eof;

my %members;
my @items = $tree->findnodes('//ul[@id="members-list"]/li');
for my $item (@items) {
    my ($member_link) = $item->findnodes('div/div[@class="item-title"]
+/a');
    my $member = $member_link->as_text;
    my ($avatar_img) = $item->findnodes('div[@class="item-avatar"]/a/i
+mg');
    my $avatar = $avatar_img->attr('src');
    my ($phone_icon) = $item->findnodes('div//i[@class="fa fa fa-phone
+"]');
    my $phone = $phone_icon->parent->as_text;
    my ($mobile_icon) = $item->findnodes('div//i[@class="fa fa fa-mobi
+le-phone"]');
    my $mobile = $mobile_icon->parent->as_text;

    $members{$member} = {
        'avatar_url'             => $avatar,
        'fa fa fa-phone'         => $phone,
        'fa fa fa-mobile-phone'  => $mobile,
    };
}
print Dumper(\%members);


__DATA__
<ul id="members-list" class="item-list" role="main">
    <li>
        <div class="item-avatar">
            <a href="https://intra.example.com/coworkers/dantealighier
+i/"><img src="SRCURL"></a>
            <span class="member-role">Sottoscrittore</span>           
+     
        </div> <!-- .item-avatar -->
        <div class="item">
            <div class="item-title">
                <a href="https://intra.example.com/coworkers/dantealig
+hieri/" class="heading"><h3>Dante Alighieri</h3></a>                 
+       
            </div>
                <div class="item-meta"><span class="activity">active 6
+ days ago, 19 hours ago</span></div>
                <div class="woffice-xprofile-list">
                    <span><i class="fa fa fa-phone"></i>011111111</spa
+n>
                    <span><i class="fa fa fa-mobile-phone"></i>3333333
+33</span>
                    <span><i class="fa fa fa-envelope-o"></i>dante.ali
+ghieri@example.com</span>
                    <span><i class="fa fa fa-check"></i>Poets and Writ
+ers</span>
                </div>                
            </div>
        <div class="action"></div>
        <div class="clear"></div>
    </li>        
    
    <li>
        <div class="item-avatar">
            <a href="https://intra.example.com/coworkers/francescopetr
+arca/"><img src="SRCURL"></a>
            <span class="member-role">Sottoscrittore</span>           
+     
        </div> <!-- .item-avatar -->
        <div class="item">
            <div class="item-title">
                <a href="https://intra.example.com/coworkers/francesco
+ptetrarca/" class="heading"><h3>Francesco Petrarca</h3></a>          
+              
            </div>
                <div class="item-meta"><span class="activity">active 7
+ days ago, 22 hours ago</span></div>
                <div class="woffice-xprofile-list">
                    <span><i class="fa fa fa-phone"></i>02222222</span
+>
                    <span><i class="fa fa fa-mobile-phone"></i></span>
                    <span><i class="fa fa fa-envelope-o"></i>francesco
+.petrarca@example.com</span>
                    <span><i class="fa fa fa-check"></i>Poets and Writ
+ers</span>
                </div>                
            </div>
        <div class="action"></div>
        <div class="clear"></div>
    </li>        
</ul>
[download]

Output:

$VAR1 = {
    'Dante Alighieri' => {
        'avatar_url'            => 'SRCURL',
        'fa fa fa-phone'        => '011111111'
        'fa fa fa-mobile-phone' => '333333333'
    },
    'Francesco Petrarca' => {
        'avatar_url'            => 'SRCURL'
        'fa fa fa-phone'        => '02222222',
        'fa fa fa-mobile-phone' => ''
    }
};
[download]